First, let us verify the ergodic capacity performance with imperfect CSI. Specifically, we consider a simple simulation setting as shown in Fig. REF , where an IRS is deployed at the location \(\left( {{d_0},0,{d_h}} \right)\) , with the horizontal BS-IRS distance of \({d_0} = 50\)  m. The multiple-IRS scenarios will be verified later by the proposed pilot power allocation scheme. We consider that the user moves far away from the IRS along the line of \(\left( {{d_0},d,0} \right)\) , where the horizontal distance between the IRS and user is increased from \(d = 4\)  m to \(d = 26\)  m. Fig. REF  plots the ergodic achievable rate under different setups, considering the number of reflecting elements of \(M = 100\)  and \(M = 1000\) , respectively; the uplink average pilot power of \(p = 0\)  dBm and \(p = 30\)  dBm, respectively. It can be seen from Fig. REF  that the achievable rate will be improved with the increase of the number of reflecting elements or the average pilot power. In all cases, our theoretical analysis is matched with the simulation results perfectly. In addition, one may note that, as the number of reflecting elements increases from \(M = 100\)  to \(M = 1000\) , the ergodic capacity is improved by about \(6.6\)  b/s/Hz, which is consistent with the quadratic power law in [1]}, i.e., \(2{\log _2}10 \approx 6.6\)  b/s/Hz. Furthermore, we note that for the same number of reflecting elements, the performance improvement caused by the increased average pilot power becomes significant as the user moves far away from the IRS. For example, the rate improvement is about 3 b/s/Hz when increasing the average pilot power from \(p = 0\)  dBm to \(p = 30\)  dBm for the user at the location of \(d = 26\)  m. Since the path loss becomes severe as the user moves far away from the IRS, it is suggested to increase the average pilot power to obtain better channel estimates and thus facilitate the coherent superposition between multiple reflection channels for the user far away from the IRS.
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